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Let $\mathrm{\λ}$ be a horizontal or vertical axis of rotation in the same plane as a plane region $A$, no interior point of which is on $\mathrm{\λ}$. A solid of revolution is formed when $A$ is rotated about $\mathrm{\λ}$.
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The volume of the solid of revolution can be found with definite integrals formed by slicing the solid either into disks or cylindrical shells. Table 5.2.1 lists the types of definite integrals that arise in the calculation of the volume of a solid of revolution.
Method

Axis

Integral

Comments

Disks

Horizontal
Vertical

$\mathrm{\π}{\int}_{a}^{b}{\mathrm{rho;}}^{2}\left(x\right)\mathit{DifferentialD;}x$
$\mathrm{\π}{\int}_{c}^{d}{\mathrm{rho;}}^{2}\left(y\right)\mathit{DifferentialD;}y$

$\mathrm{\ρ}$ is the (varying) radius of rotation, the distance from the axis of rotation $\mathrm{\λ}$, to the boundary of the rotated region $A$.

Washers
(Disks with holes)

Horizontal
Vertical

$\mathrm{\π}{\int}_{a}^{b}\left({R}^{2}\left(x\right){r}^{2}\left(x\right)\right)\mathit{DifferentialD;}x$
$\mathrm{\π}{\int}_{c}^{d}\left({R}^{2}\left(y\right){r}^{2}\left(y\right)\right)\mathit{DifferentialD;}y$

$R$ is the radius of rotation of the outer boundary of $A$, whereas $r$ is the radius of rotation of the inner boundary.

Shells

Vertical
Horizontal

$2\mathrm{pi;}{\int}_{a}^{b}\mathrm{rho;}\left(x\right)L\left(x\right)\mathit{DifferentialD;}x$
$2\mathrm{pi;}{\int}_{c}^{d}\mathrm{rho;}\left(y\right)L\left(y\right)\mathit{DifferentialD;}y$

$\mathrm{\ρ}$ is the (varying) radius of the shells.
$L$ is the (varying) length of a cylindrical shell.

Table 5.2.1 The methods of disks and shells for calculating the volume of a solid of revolution



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